Calculating standard form without a calculator
Adding and subtracting
When adding and subtracting standard formA system in which numbers are written as a number greater than 1 and less than 10 multiplied by a power of 10 which may be positive or negative. numbers, an easy way is to:
- convert the numbers from standard form into decimal form or ordinary numbers
- complete the calculation
- convert the answer back into standard form
Example
Calculate \((4.5 \times 10^4) + (6.45 \times 10^6)\).
\(= 45,000 + 6,450,000\)
\(= 6,495,000\)
\(= 6.495 \times 10^6\)
Question
Calculate \((8.5 \times 10^7) - (1.23 \times 10^4)\).
\(= 85,000,000 - 12,300\)
\(= 84,987,700\)
\(= 8.49877 \times 10^7\)
Multiplying and dividing
When multiplying and dividing you can use the Laws of Indices:
- multiply or divide the first numbers
- apply the Laws of Indices to the powers of 10
Example 1
Calculate \((3 \times 10^3) \times (3 \times 10^9)\).
Multiply the first numbers – which in this case is \(3 \times 3 = 9\).
Apply the index law on the powers of 10:
- \(10^3 \times 10^9 = 10^{3 + 9} = 10^{12}\)
- \((3 \times 10^3) \times (3 \times 10^9) = 9 \times 10^{12}\)
Take care that the answer is in standard form. It is common to have to re-adjust the answer.
Example 2
Calculate \((4 \times 10^9) \times (7 \times 10^{-3})\).
Multiply the first numbers \(4 \times 7 = 28\).
Apply the Laws of Indices on the powers:
- \(10^9 \times 10^{-3} = 10^{9 + -3} = 10^6\)
- \((4 \times 10^9) \times (7 \times 10^{-3}) = 28 \times 10^6\)
But \(28 \times 10^6\) is not in standard form, as the first number is not between 1 and 10. To correct this, divide 28 by 10 so that it is a number between 1 and 10. To balance out that division of 10, multiply the second part by 10 which gives 107.
\(28 \times 10^6\) and \(2.8 \times 10^7\) are equivalent but only the second is written in standard form.
So, \((4 \times 10^9) \times (7 \times 10^{-3}) = 2.8 \times 10^7\)
Question
Calculate \((2 \times 10^7) \div (8 \times 10^2)\).
- Divide the first numbers: \(2 \div 8 = 0.25\)
- Apply the Laws of Indices on the powers
- \(10^7 \div 10^2 = 10^{7 - 2} = 10^5\)
- So, \((2 \times 10^7) \div (8 \times 10^2) = 0.25 \times 10^5\)
But \(0.25 \times 10^5\) is not in standard form as the first number is not between 1 and 10. To correct this, multiply 0.25 by 10 so that it is a number between 1 and 10. To balance out that multiplication of 10, divide the second part by 10 which gives 104. So \(0.25 \times 10^5\) and \(2.5 \times 10^4\) are equivalent but only the second is written in standard form.
So, \((2 \times 10^7) \div (8 \times 10^2) = 2.5 \times 10^4\)